\(\int \frac {A+B x^2}{(d+e x^2)^2 (a+c x^4)^{3/2}} \, dx\) [79]

Optimal result

Integrand size = 28, antiderivative size = 830 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=-\frac {e (B d-A e) x}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right ) \sqrt {a+c x^4}}+\frac {c x \left (d \left (A c d^2+3 a B d e-2 a A e^2\right )+\left (B c d^3-2 A c d^2 e-2 a B d e^2+a A e^3\right ) x^2\right )}{2 a d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt {c} \left (B c d^3-2 A c d^2 e-2 a B d e^2+a A e^3\right ) x \sqrt {a+c x^4}}{2 a d \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^{3/2} \left (5 B c d^3-7 A c d^2 e-a B d e^2-a A e^3\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}+\frac {\sqrt [4]{c} \left (B c d^3-2 A c d^2 e-2 a B d e^2+a A e^3\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (A c d^2-a e (B d-2 A e)-\sqrt {a} \sqrt {c} d (B d-A e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (5 B c d^3-7 A c d^2 e-a B d e^2-a A e^3\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \] Output:

-1/2*e*(-A*e+B*d)*x/d/(a*e^2+c*d^2)/(e*x^2+d)/(c*x^4+a)^(1/2)+1/2*c*x*(d*( 
-2*A*a*e^2+A*c*d^2+3*B*a*d*e)+(A*a*e^3-2*A*c*d^2*e-2*B*a*d*e^2+B*c*d^3)*x^ 
2)/a/d/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-1/2*c^(1/2)*(A*a*e^3-2*A*c*d^2*e-2* 
B*a*d*e^2+B*c*d^3)*x*(c*x^4+a)^(1/2)/a/d/(a*e^2+c*d^2)^2/(a^(1/2)+c^(1/2)* 
x^2)-1/4*e^(3/2)*(-A*a*e^3-7*A*c*d^2*e-B*a*d*e^2+5*B*c*d^3)*arctan((a*e^2+ 
c*d^2)^(1/2)*x/d^(1/2)/e^(1/2)/(c*x^4+a)^(1/2))/d^(3/2)/(a*e^2+c*d^2)^(5/2 
)+1/2*c^(1/4)*(A*a*e^3-2*A*c*d^2*e-2*B*a*d*e^2+B*c*d^3)*(a^(1/2)+c^(1/2)*x 
^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/ 
4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/d/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)+1/4* 
c^(1/4)*(A*c*d^2-a*e*(-2*A*e+B*d)-a^(1/2)*c^(1/2)*d*(-A*e+B*d))*(a^(1/2)+c 
^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*ar 
ctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(5/4)/d/(c^(1/2)*d-a^(1/2)*e)/(a*e^ 
2+c*d^2)/(c*x^4+a)^(1/2)+1/8*e*(c^(1/2)*d+a^(1/2)*e)*(-A*a*e^3-7*A*c*d^2*e 
-B*a*d*e^2+5*B*c*d^3)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^ 
2)^2)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(1/2)*d-a^ 
(1/2)*e)^2/a^(1/2)/c^(1/2)/d/e,1/2*2^(1/2))/a^(1/4)/c^(1/4)/d^2/(c^(1/2)*d 
-a^(1/2)*e)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.67 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.51 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d \left (a e^3 (-B d+A e) x \left (a+c x^4\right )+c d x \left (d+e x^2\right ) \left (-a A e^2+B c d^2 x^2+A c d \left (d-2 e x^2\right )+a B e \left (2 d-e x^2\right )\right )\right )-\left (d+e x^2\right ) \sqrt {1+\frac {c x^4}{a}} \left (-\sqrt {a} \sqrt {c} d \left (-B c d^3+2 A c d^2 e+2 a B d e^2-a A e^3\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) \left (A c d^2+i \sqrt {a} \sqrt {c} d (B d-A e)+a e (2 B d-A e)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+a e \left (-5 B c d^3+7 A c d^2 e+a B d e^2+a A e^3\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (c d^3+a d e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+c x^4}} \] Input:

Integrate[(A + B*x^2)/((d + e*x^2)^2*(a + c*x^4)^(3/2)),x]
 

Output:

(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*(a*e^3*(-(B*d) + A*e)*x*(a + c*x^4) + c*d*x*( 
d + e*x^2)*(-(a*A*e^2) + B*c*d^2*x^2 + A*c*d*(d - 2*e*x^2) + a*B*e*(2*d - 
e*x^2))) - (d + e*x^2)*Sqrt[1 + (c*x^4)/a]*(-(Sqrt[a]*Sqrt[c]*d*(-(B*c*d^3 
) + 2*A*c*d^2*e + 2*a*B*d*e^2 - a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[ 
c])/Sqrt[a]]*x], -1]) + I*(Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(A*c*d^2 + 
I*Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(2*B*d - A*e))*EllipticF[I*ArcSinh[S 
qrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + a*e*(-5*B*c*d^3 + 7*A*c*d^2*e + a*B*d*e 
^2 + a*A*e^3)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*S 
qrt[c])/Sqrt[a]]*x], -1])))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(c*d^3 + a*d*e^ 
2)^2*(d + e*x^2)*Sqrt[a + c*x^4])
 

Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 1494, normalized size of antiderivative = 1.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2259, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x^2}{\left (a+c x^4\right )^{3/2} \left (d+e x^2\right )^2} \, dx\)

\(\Big \downarrow \) 2259

\(\displaystyle \int \left (\frac {e (A e-B d)}{\sqrt {a+c x^4} \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}+\frac {e \left (a B e^2+2 A c d e-B c d^2\right )}{\sqrt {a+c x^4} \left (d+e x^2\right ) \left (a e^2+c d^2\right )^2}+\frac {c \left (x^2 \left (-a B e^2-2 A c d e+B c d^2\right )-a A e^2+2 a B d e+A c d^2\right )}{\left (a+c x^4\right )^{3/2} \left (a e^2+c d^2\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(B d-A e) x \sqrt {c x^4+a} e^3}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e^2}{2 d \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt {c} (B d-A e) x \sqrt {c x^4+a} e^2}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {(B d-A e) \left (3 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (B c d^2-2 A c e d-a B e^2\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {c x^4+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B e d-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt {c} \left (B c d^2-2 A c e d-a B e^2\right ) x \sqrt {c x^4+a}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {c x \left (A c d^2+2 a B e d-a A e^2+\left (B c d^2-2 A c e d-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}\)

Input:

Int[(A + B*x^2)/((d + e*x^2)^2*(a + c*x^4)^(3/2)),x]
 

Output:

(c*x*(A*c*d^2 + 2*a*B*d*e - a*A*e^2 + (B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x^2) 
)/(2*a*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqr 
t[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (Sqrt[c]*( 
B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)^2*( 
Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 
+ a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 + a*e^2)*ArcTan[(S 
qrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*(c*d^ 
2 + a*e^2)^(5/2)) - (e^(3/2)*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*ArcTan[(Sqrt[ 
c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*d^2 + 
a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*S 
qrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/ 
a^(1/4)], 1/2])/(2*d*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)*(B*c*d^ 
2 - 2*A*c*d*e - a*B*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] 
 + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/ 
4)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + 
Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTa 
n[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 
+ a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*(Sq 
rt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF 
[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2259

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] 
 :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p 
+ 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 
2] && IntegerQ[q]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.90 (sec) , antiderivative size = 1384, normalized size of antiderivative = 1.67

method	result	size
default	\(\text {Expression too large to display}\)	\(1384\)
elliptic	\(\text {Expression too large to display}\)	\(1664\)

Input:

int((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

B/e*(-2*c*(1/4/a*e/(a*e^2+c*d^2)*x^3-1/4/a*d/(a*e^2+c*d^2)*x)/(c*(a/c+x^4) 
)^(1/2)+1/2*c/a*d/(a*e^2+c*d^2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2 
/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF( 
x*(I*c^(1/2)/a^(1/2))^(1/2),I)+1/2*I*c^(1/2)/a^(1/2)*e/(a*e^2+c*d^2)/(I*c^ 
(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1 
/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-1/2*I* 
c^(1/2)/a^(1/2)*e/(a*e^2+c*d^2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2 
/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticE( 
x*(I*c^(1/2)/a^(1/2))^(1/2),I)+1/(a*e^2+c*d^2)*e^2/d/(I*c^(1/2)/a^(1/2))^( 
1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^ 
4+a)^(1/2)*EllipticPi(x*(I*c^(1/2)/a^(1/2))^(1/2),I/c^(1/2)*a^(1/2)/d*e,(- 
I/a^(1/2)*c^(1/2))^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)))+(A*e-B*d)/e*(1/2*e^4/ 
(a*e^2+c*d^2)^2/d*x*(c*x^4+a)^(1/2)/(e*x^2+d)-2*c*(1/2/a*d*e*c/(a*e^2+c*d^ 
2)^2*x^3+1/4/a*(a*e^2-c*d^2)/(a*e^2+c*d^2)^2*x)/(c*(a/c+x^4))^(1/2)-1/(I*c 
^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^( 
1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)*e^2*c 
/(a*e^2+c*d^2)^2+1/2/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^( 
1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2 
)/a^(1/2))^(1/2),I)*c^2/a/(a*e^2+c*d^2)^2*d^2-1/2*I*a^(1/2)/(I*c^(1/2)/a^( 
1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {A + B x^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{2}}\, dx \] Input:

integrate((B*x**2+A)/(e*x**2+d)**2/(c*x**4+a)**(3/2),x)
 

Output:

Integral((A + B*x**2)/((a + c*x**4)**(3/2)*(d + e*x**2)**2), x)
 

Maxima [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)/((c*x^4 + a)^(3/2)*(e*x^2 + d)^2), x)
 

Giac [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)/((c*x^4 + a)^(3/2)*(e*x^2 + d)^2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (c\,x^4+a\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:

int((A + B*x^2)/((a + c*x^4)^(3/2)*(d + e*x^2)^2),x)
 

Output:

int((A + B*x^2)/((a + c*x^4)^(3/2)*(d + e*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} e^{2} x^{12}+2 c^{2} d e \,x^{10}+2 a c \,e^{2} x^{8}+c^{2} d^{2} x^{8}+4 a c d e \,x^{6}+a^{2} e^{2} x^{4}+2 a c \,d^{2} x^{4}+2 a^{2} d e \,x^{2}+a^{2} d^{2}}d x \right ) a +\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} e^{2} x^{12}+2 c^{2} d e \,x^{10}+2 a c \,e^{2} x^{8}+c^{2} d^{2} x^{8}+4 a c d e \,x^{6}+a^{2} e^{2} x^{4}+2 a c \,d^{2} x^{4}+2 a^{2} d e \,x^{2}+a^{2} d^{2}}d x \right ) b \] Input:

int((B*x^2+A)/(e*x^2+d)^2/(c*x^4+a)^(3/2),x)
 

Output:

int(sqrt(a + c*x**4)/(a**2*d**2 + 2*a**2*d*e*x**2 + a**2*e**2*x**4 + 2*a*c 
*d**2*x**4 + 4*a*c*d*e*x**6 + 2*a*c*e**2*x**8 + c**2*d**2*x**8 + 2*c**2*d* 
e*x**10 + c**2*e**2*x**12),x)*a + int((sqrt(a + c*x**4)*x**2)/(a**2*d**2 + 
 2*a**2*d*e*x**2 + a**2*e**2*x**4 + 2*a*c*d**2*x**4 + 4*a*c*d*e*x**6 + 2*a 
*c*e**2*x**8 + c**2*d**2*x**8 + 2*c**2*d*e*x**10 + c**2*e**2*x**12),x)*b